A wide variety of plasma flows can be found in the various planetary ionospheres. For example, gentle near-equilibrium flows occur in the terrestrial ionosphere at mid-latitudes, while highly nonequilibrium flow conditions exist in the terrestrial polar wind and in the Venus ionosphere near the solar terminator. The highly nonequilibrium flows are generally characterized by large temperature differences between the interacting species, by flow speeds approaching and exceeding thermal speeds, and by flow conditions changing from collision-dominated to collisionless regimes. In an effort to model the various ionospheric flow conditions, several different mathematical approaches have been used, including collision-dominated and collisionless transport equations, kinetic and semikinetic models, and macroscopic particle-in-cell techniques. However, the transport equation approach has received the most attention, primarily because it can handle most of the flow conditions encountered in planetary ionospheres. Therefore, the main focus of this chapter is on transport theory, although other mathematical approaches are briefly discussed at the end of the chapter. Typically, numerous assumptions are made to simplify the transport equations before they are applied, and therefore, it is instructive to trace the derivation of the various sets of transport equations in order to establish their intrinsic strengths and limitations. Before diving into the rigorous derivation of the transport equations, it is useful to review the simple derivation of the continuity equation given in Appendix N.

Boltzmann equation

The Boltzmann equation is not only the starting point for the derivation of the different sets of transport equations but also forms the basis for the kinetic and semikinetic theories. With Boltzmann's approach, one is not interested in the motion of individual particles in the gas, but instead with the distribution of particles.